Short answer: do not take it literally, without further context.
In order to understand the Higgs boson's role in the Standard model, it is necessary to take a closer look at the framework in which we describe elementary particles: quantum field theory.
In this approach, particles are described as excitations of fields that spans all spacetime. The ground state of the field corresponds to vacuum, what we call particles corresponds to excitations of the latter. If you are familiar with quantum mechanics, think of a harmonic oscillator to understand the concept.
The point is now that the mass creation effect is due to the presence of the field, not the associated particle. The existence of the particle arises as a kind of consistency requirement, which was confirmed recently at the LHC.
In this sense, the answer to your question depends entirely on what you mean by "Higgs boson": having massive particles does not mean that there are bosons in the sense of particles constantly surrounding them. They are far too heavy for this to be a viable option, as 125 GeV is far beyond what one experiences in daily life (a proton comes at a rest mass of almost 1 GeV).
For a particle physicist, it is obvious that "boson" refers to a property of the field as a whole, and not the excitation of the latter. A layman, however, will associate it with a particle moving through space time. Hence, you should not take this phrase literally. I would refrain from using it freely without providing any additional explanation.